No; not all contracts need to be in writing. In some cases, statute law requires contracts to be in writing (e.g. contracts to buy and sell land, to buy a car, and door-to-door sales contracts). However, it is always useful to write down the terms agreed by the parties so later on there is less potential for dispute about what was agreed. A written contract can be drawn up by listing all the terms agreed by the parties and getting each party to sign and date the document.
It is useful to keep any documents relating to the agreement (e.g. quotations, invoices and receipts), so if a dispute does arise, these documents can assist to resolve differences between the parties.